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Ιακωβιανή
Ιακωβιανή Jacobian 600px|thumb|center| [[Μετασχηματισμός Ιακωβιανή ]] thumb|300px| [[Μετασχηματισμός Ιακωβιανή ]] thumb|300px| [[Επιφάνεια ΚαμπυλότηταΙακωβιανή ]] thumb|300px| [[Ιακωβιανή ]] thumb|300px| [[Ιακωβιανή ]] thumb|300px| [[Ιακωβιανή ]] thumb|300px| [[Ιακωβιανή ]] thumb|300px| [[Ιακωβιανή ]] thumb|300px| [[Ιακωβιανή ]] thumb|300px| [[Ιακωβιανή ]] thumb|300px| [[Ιακωβιανή Χωροπεριοχή ]] - Μία Μαθηματική Μήτρα. Ετυμολογία Η ονομασία "Ιακωβιανή" σχετίζεται ετυμολογικά με την λέξη " ". Εισαγωγή Στον Διανυσματικό Λογισμό, the Jacobian is shorthand for either the Jacobian matrix or its [ορίζουσα (determinant), the Jacobian determinant. Also, in Αλγεβρική Γεωμετρία (algebraic geometry) the Jacobian of a curve means the Jacobian variety: a group variety associated to the curve, in which the καμπύλη (curve) can be embedded. They are all named after the mathematician Carl Gustav Jacobi; the term "Jacobian" is pronounced but is very often pronounced among English speakers. Jacobian matrix The Jacobian matrix is the Μαθηματική Μήτρα of all first-order partial derivatives of a vector-valued function. Its importance lies in the fact that it represents the best γραμμική προσέγγιση to a differentiable function near a given point. In this sense, the Jacobian is akin to a derivative of a multivariate function. Suppose F'' : '''Rn'' → '''Rm'' is a function from [[Euclidean space|Euclidean ''n-space]] to Euclidean m''-space. Such a function is given by ''m real-valued component functions, y''1(''x''1,...,''xn''), ..., ''ym''(''x''1,...,''xn''). The partial derivatives of all these functions (if they exist) can be organized in an ''m-by-''n'' matrix, the Jacobian matrix of F'', as follows: : \begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \cdots & \frac{\partial y_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial y_m}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_n} \end{bmatrix} This matrix is denoted by : J_F(x_1,\ldots,x_n) or by \frac{\partial(y_1,\ldots,y_m)}{\partial(x_1,\ldots,x_n)} The ''i''th row of this matrix is given by the gradient of the function ''yi'' for ''i=1,...,m''. If '''p' is a point in R''n'' and F'' is differentiable at '''p', then its derivative is given by JF(p') (and this is the easiest way to compute the derivative). In this case, the linear map described by ''JF('''p) is the best linear approximation of F'' near the point '''p', in the sense that : F(\mathbf{x}) \approx F(\mathbf{p}) + J_F(\mathbf{p})\cdot (\mathbf{x}-\mathbf{p}) for x''' close to '''p. Παραδείγματα Consider Σφαιρικές Συντεταγμένες (spherical polar coordinates). The following function constitutes a change of variables back to cartesian coordinates. F'' : '''R' x 0,2π x 0,π → R'3 with components: : x_1 = r \cos\phi \sin\theta \, : x_2 = r \sin\phi \sin\theta \, : x_3 = r \cos\theta \, The Jacobian matrix for this coordinate change is : J_F(r,\phi,\theta) =\begin{bmatrix} \cos\phi \sin\theta & -r \sin\phi \sin\theta & r \cos\phi \cos\theta \\ \sin\phi \sin\theta & r \cos\phi \sin\theta & r \sin\phi \cos\theta \\ \cos\theta & 0 & -r \sin\theta \end{bmatrix} The Jacobian matrix of the function ''F : 'R'3 → 'R'4 with components: : y_1 = x_1 \, : y_2 = 5x_3 \, : y_3 = 4x_2^2 - 2x_3 \, : y_4 = x_3 \sin(x_1) \, είναι: : J_F(x_1,x_2,x_3) =\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 5 \\ 0 & 8x_2 & -2 \\ x_3\cos(x_1) & 0 & \sin(x_1) \end{bmatrix} This example shows that the Jacobian need not be a square matrix. Ιακωβιανή Δυναμικών Συστημάτων Consider a dynamical system of the form ''x = F''(''x), with F'' : '''Rn'' → '''Rn''. If ''F(x''0) = 0, then ''x''0 is a stationary point. The behaviour of the system near a stationary point can often be determined by the eigenvalues of ''JF''(''x''0), the Jacobian of ''F at the stationary point.D.K. Arrowsmith and C.M. Place, Dynamical Systems, Section 3.3, Chapman & Hall, London, 1992. ISBN 0-412-39080-9. Jacobian determinant If m'' = ''n, then F'' is a function from ''n-space to n''-space and the Jacobi matrix is a square matrix. We can then form its determinant, known as the '''Jacobian determinant'. The Jacobian determinant is also called the "Jacobian" in some sources. The Jacobian determinant at a given point gives important information about the behavior of F'' near that point. For instance, the continuously differentiable function ''F is invertible near p''' if the Jacobian determinant at '''p is non-zero. This is the inverse function theorem. Furthermore, if the Jacobian determinant at p''' is positive, then F preserves orientation near '''p; if it is negative, F'' reverses orientation. The absolute value of the Jacobian determinant at '''p' gives us the factor by which the function F'' expands or shrinks volumes near '''p'; this is why it occurs in the general substitution rule. Example The Jacobian determinant of the function F'' : '''R'3 → 'R'3 with components : y_1 = 5x_2 \, : y_2 = 4x_1^2 - 2 \sin (x_2x_3) \, : y_3 = x_2 x_3 \, is: : \begin{vmatrix} 0 & 5 & 0 \\ 8x_1 & -2x_3\cos(x_2 x_3) & -2x_2\cos(x_2 x_3) \\ 0 & x_3 & x_2 \end{vmatrix}=-8x_1\cdot\begin{vmatrix} 5 & 0\\ x_3&x_2\end{vmatrix}=-40x_1 x_2 From this we see that F'' reverses orientation near those points where ''x''1 and ''x''2 have the same sign; the function is locally invertible everywhere except near points where ''x''1=0 or ''x''2=0. If you start with a tiny object around the point (1,1,1) and apply ''F to that object, you will get an object set with about 40 times the volume of the original one. Uses The Jacobian determinant is used when making a change of variables when integrating a function over its domain. To accommodate for the change of coordinates the Jacobian determinant arises as a multiplicative factor within the integral. Normally it is required that the change of coordinates is done in a manner which maintains an injectivity between the coordinates that determine the domain. The Jacobian determinant, as a result, is usually well defined. See also * Push forward * Hessian matrix References Εσωτερική Αρθρογραφία * συνάρτηση * μετασχηματισμός Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] Αγγλική Ιστογραφία * Ian Craw's Undergraduate Teaching Page An easy to understand explanation of Jacobians * Mathworld A more technical explanation of Jacobians Category: Συναρτήσεις Category: Μαθηματικές Μήτρες Category: Ορίζουσες